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Mirrors > Home > ILE Home > Th. List > fnop | GIF version |
Description: The first argument of an ordered pair in a function belongs to the function's domain. (Contributed by NM, 8-Aug-1994.) |
Ref | Expression |
---|---|
fnop | ⊢ ((𝐹 Fn 𝐴 ∧ 〈𝐵, 𝐶〉 ∈ 𝐹) → 𝐵 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3988 | . 2 ⊢ (𝐵𝐹𝐶 ↔ 〈𝐵, 𝐶〉 ∈ 𝐹) | |
2 | fnbr 5298 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ 𝐴) | |
3 | 1, 2 | sylan2br 286 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 〈𝐵, 𝐶〉 ∈ 𝐹) → 𝐵 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2141 〈cop 3584 class class class wbr 3987 Fn wfn 5191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-xp 4615 df-rel 4616 df-dm 4619 df-fun 5198 df-fn 5199 |
This theorem is referenced by: 2elresin 5307 tfrlem9 6296 tfrexlem 6311 |
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